How to Prove the Diagonals of a Rhombus Are Perpendicular

TL;DR

To prove the diagonals of a rhombus are perpendicular, demonstrate that the triangles formed by the diagonals are congruent using side-side-side congruency. Since the corresponding angles are congruent and supplementary, each angle measures 90 degrees, resulting in the diagonals intersecting at right angles.

Transcript

We're told that quadrilateral ABCD is a rhombus. And what they want us to prove is that their diagonals are perpendicular, that AC is perpendicular to BD. Now let's think about everything we know about a rhombus. First of all, a rhombus is a special case of a parallelogram. In a parallelogram, the opposite sides are parallel. So that side is parall... Read More

Key Insights

• 🙃 A rhombus is a special type of parallelogram with equal sides and parallel opposite sides.
• 🫤 The diagonals in a parallelogram bisect each other.
• 🔺 By proving congruent triangles and corresponding angles in a rhombus, the perpendicularity of its diagonals can be established.
• 🫤 The diagonals of a rhombus intersect at a right angle, forming perpendicular bisectors.
• 🫤 The properties of a rhombus and parallelogram can be used to prove the perpendicularity of its diagonals.
• 🔺 Angle measures and congruent triangles are important tools in proving properties of geometric shapes.
• 🫤 The diagonals in a rhombus divide it into four congruent right triangles.

Questions & Answers

Q: What is a rhombus, and what properties does it have?

A rhombus is a special type of parallelogram with all sides equal in length. Its opposite sides are parallel, and all its angles are equal.

Q: How do diagonals in a parallelogram bisect each other?

Diagonals in a parallelogram bisect each other, meaning they intersect at a point that divides each diagonal into two equal parts.

Q: How can congruent triangles help in proving perpendicular diagonals in a rhombus?

By proving that two triangles in a rhombus are congruent, it can be shown that corresponding angles in those triangles are also congruent. If two angles are congruent and supplementary, they must be right angles.

Q: Why are the diagonals in a rhombus perpendicular?

The congruent triangles and corresponding angles in a rhombus prove that the measure of angle AEB is equal to the measure of angle CEB, which is equal to 90 degrees. Hence, the diagonals in a rhombus are perpendicular.

Summary & Key Takeaways

• A rhombus is a special case of a parallelogram with all sides equal in length and opposite sides parallel.

• Diagonals in a parallelogram bisect each other, meaning they divide each other into two equal parts.

• By proving congruent triangles and corresponding angles in a rhombus, it can be shown that the diagonals are perpendicular.